How can I run a add-on from a python script
Imagine there is an addon installed called "my addon". It has a couple properties to control in the toolshelf, and an operator. 

The properties are all in bpy.context.scene. 

In your script/addon, you can access all these properties. 

You can set properties by

bpy.context.scene.my_addon_string = "hey"


And call operators

bpy.ops.object.my_addon_op()


So with a combination of editing properties and running operators, you should be able to run the addon from another addon or script.